7.3 Scientific Significance of Flux Pinning Phenomena
169
the dynamic frictional force caused by its motion in the potential field is expected to
approach zero asymptotically, as shown by the chained line in Fig. 7.7.
3 In practice,
energy dissipation occurs due to the elastic damped oscillations of the projections on
each rough surface. These two factors result in the dynamic frictional force, which
diminishes but approaches a finite value, as illustrated in Fig. 7.7. In the case of flux
pinning in a superconductor, the flux line lattice behaves elastically, and the resultant
dynamic pinning force is similar to the static pinning force. This is assumed to be
the same in the dynamic critical state model.
The common point in these phenomena is random interactions between a moving
body and the matter that restricts the motion, which can be described as a potential.
These interactions are reversible when the displacement of a flux line, magnetic
domain wall, or body is small enough. In particular, such reversible phenomena can be
measured when a variation in magnetization is observed due to a small AC magnetic
field on a major magnetization curve. The play of a body when it is displaced slightly
by a force corresponds to this in a mechanical system. The reversibility shows that
such interactions are caused by potential, and this is the essence of those phenomena
that show hysteresis under large displacements.
In the case of electromagnetic phenomena in a superconductor, the irreversibility
was proved by the summation theory of flux pinning. The irreversibility does not
come from the breaking of time reversal symmetry of the equation of motion but
from the change in the force balance equation before and after the change in the
direction of motion due to unstable flux motion. The unstable flux motion is also
responsible for the hysteresis loss through the mechanism of viscous loss. A similar
theoretical system can be constructed for the motion of magnetic domain walls in
ferromagnetic materials or for friction in mechanical systems. The irreversibility in
these systems is also derived from the change in the governing equation due to the
instability. Hence, this kind of theory can be generalized to describe irreversibility
of this nature, although some correction is necessary. In the case of the motion of
magnetic domain walls, the spacing between magnetic domain walls is wide, and
there is no periodicity among them, so a model of isolated pinning centers seems to
be adequate instead of the periodic pinning model used in this book. It seems to be
necessary to assume irregular pinning centers with high density for friction.
The longitudinal magnetic field effect was introduced as a new electromagnetic
phenomenon in this book. The force-free state that appears in the longitudinal
magnetic field effect is a special state with finite magnetic helicity in the static state,
and hence, it can be achieved only in superconductors. The current and magnetic
flux lines are locally parallel to each other, and the resultant Lorentz force is zero
in this state. There is a characteristic torsion, however, i.e., a rotational shearing
torsion, in the flux line system in this state, and a torque works on flux lines to reduce
the torsion. This is the force-free torque. The pinning energy is shared between
the force-balance and the torque balance. The flux line system is self-organized to
3 See J. E. Evetts and J. R. Appleyard: Proc. Int. Disc. Meeting on Flux Pinning in Supercond.,
Göttingen, 1974, p. 69, or T. Matsushita, “Flux Pinning in Superconductors, Ed. 2,” (Springer,
2014) p. 336 (Exercise 7.4).
169
the dynamic frictional force caused by its motion in the potential field is expected to
approach zero asymptotically, as shown by the chained line in Fig. 7.7.
3 In practice,
energy dissipation occurs due to the elastic damped oscillations of the projections on
each rough surface. These two factors result in the dynamic frictional force, which
diminishes but approaches a finite value, as illustrated in Fig. 7.7. In the case of flux
pinning in a superconductor, the flux line lattice behaves elastically, and the resultant
dynamic pinning force is similar to the static pinning force. This is assumed to be
the same in the dynamic critical state model.
The common point in these phenomena is random interactions between a moving
body and the matter that restricts the motion, which can be described as a potential.
These interactions are reversible when the displacement of a flux line, magnetic
domain wall, or body is small enough. In particular, such reversible phenomena can be
measured when a variation in magnetization is observed due to a small AC magnetic
field on a major magnetization curve. The play of a body when it is displaced slightly
by a force corresponds to this in a mechanical system. The reversibility shows that
such interactions are caused by potential, and this is the essence of those phenomena
that show hysteresis under large displacements.
In the case of electromagnetic phenomena in a superconductor, the irreversibility
was proved by the summation theory of flux pinning. The irreversibility does not
come from the breaking of time reversal symmetry of the equation of motion but
from the change in the force balance equation before and after the change in the
direction of motion due to unstable flux motion. The unstable flux motion is also
responsible for the hysteresis loss through the mechanism of viscous loss. A similar
theoretical system can be constructed for the motion of magnetic domain walls in
ferromagnetic materials or for friction in mechanical systems. The irreversibility in
these systems is also derived from the change in the governing equation due to the
instability. Hence, this kind of theory can be generalized to describe irreversibility
of this nature, although some correction is necessary. In the case of the motion of
magnetic domain walls, the spacing between magnetic domain walls is wide, and
there is no periodicity among them, so a model of isolated pinning centers seems to
be adequate instead of the periodic pinning model used in this book. It seems to be
necessary to assume irregular pinning centers with high density for friction.
The longitudinal magnetic field effect was introduced as a new electromagnetic
phenomenon in this book. The force-free state that appears in the longitudinal
magnetic field effect is a special state with finite magnetic helicity in the static state,
and hence, it can be achieved only in superconductors. The current and magnetic
flux lines are locally parallel to each other, and the resultant Lorentz force is zero
in this state. There is a characteristic torsion, however, i.e., a rotational shearing
torsion, in the flux line system in this state, and a torque works on flux lines to reduce
the torsion. This is the force-free torque. The pinning energy is shared between
the force-balance and the torque balance. The flux line system is self-organized to
3 See J. E. Evetts and J. R. Appleyard: Proc. Int. Disc. Meeting on Flux Pinning in Supercond.,
Göttingen, 1974, p. 69, or T. Matsushita, “Flux Pinning in Superconductors, Ed. 2,” (Springer,
2014) p. 336 (Exercise 7.4).